The Sequential Quadratic Programming Method

Sequential (or Successive) Quadratic Programming (SQP) is a technique for the solution of Nonlinear Programming (NLP)problems. It is, as we shall see, an idealized concept, permitting and indeed necessitating many variations and modifications before becoming available as part of a reliable andefficient production computer code. In this monograph we trace the evolution of the SQP method through some important special cases of nonlinear programming, up to the most general form of problem. To fully understandthese developments it is important to have a thorough grasp of the underlying theoretical concepts, particularly in regard to optimality conditions. In this monograph we include a simple yet rigorous presentation of optimality conditions, which yet covers most cases of interest.

[1]  R. Courant.  Variational methods for the solution of problems of equilibrium and vibrations , 1943 .

[2]  Jerzy Neyman,et al.  Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability : held at the Statistical Laboratory, Department of Mathematics, University of California, July 31-August 12, 1950 , 1951 .

[3]  W. R. Buckland,et al.  Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. , 1952 .

[4]  O. Mangasarian,et al.  The Fritz John Necessary Optimality Conditions in the Presence of Equality and Inequality Constraints , 1967 .

[5]  M. Powell.  A method for nonlinear constraints in minimization problems , 1969 .

[6]  T. Pietrzykowski.  An Exact Potential Method for Constrained Maxima , 1969 .

[7]  M. Hestenes.  Multiplier and gradient methods , 1969 .

[8]  S. M. Robinson,et al.  A quadratically-convergent algorithm for general nonlinear programming problems , 1972, Math. Program..

[9]  J. J. Moré,et al.  A Characterization of Superlinear Convergence and its Application to Quasi-Newton Methods , 1973 .

[10]  R. Rockafellar.  Augmented Lagrange Multiplier Functions and Duality in Nonconvex Programming , 1974 .

[11]  Shih-Ping Han.  A globally convergent method for nonlinear programming , 1975 .

[12]  P. Toint.  On sparse and symmetric matrix updating subject to a linear equation , 1977 .

[13]  N. Maratos,et al.  Exact penalty function algorithms for finite dimensional and control optimization problems , 1978 .

[14]  M. J. D. Powell,et al.  A fast algorithm for nonlinearly constrained optimization calculations , 1978 .

[15]  J. Nocedal.  Updating Quasi-Newton Matrices With Limited Storage , 1980 .

[16]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[17]  R. M. Chamberlain,et al.  Algorithms for constrained minimization of smooth nonlinear functions , 1982 .

[18]  R. Fletcher.  A model algorithm for composite nondifferentiable optimization problems , 1982 .

[19]  Michael A. Saunders,et al.  A projected Lagrangian algorithm and its implementation for sparse nonlinear constraints , 1982 .

[20]  C. Lemaréchal,et al.  The watchdog technique for forcing convergence in algorithms for constrained optimization , 1982 .

[21]  R. Fletcher,et al.  Second order corrections for non-differentiable optimization , 1982 .

[22]  L. Grippo,et al.  A nonmonotone line search technique for Newton's method , 1986 .

[23]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[24]  李幼升,et al.  Ph , 1989 .

[25]  Roger Fletcher,et al.  Nonlinear programming and nonsmooth optimization by successive linear programming , 1989, Math. Program..

[26]  Roman A. Polyak,et al.  Modified barrier functions (theory and methods) , 1992, Math. Program..

[27]  Brian W. Kernighan,et al.  AMPL: A Modeling Language for Mathematical Programming , 1993 .

[28]  J. Henry,et al.  System Modelling and Optimization: Proceedings of the 16th IFIP-TC7 Conference, Compiègne, France, July 5-9, 1993 , 1994, System Modelling and Optimization.

[29]  Jorge Nocedal,et al.  Representations of quasi-Newton matrices and their use in limited memory methods , 1994, Math. Program..

[30]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[31]  Nicholas I. M. Gould,et al.  CUTE: constrained and unconstrained testing environment , 1995, TOMS.

[32]  R. Fletcher,et al.  Computing Sparse Hessian and Jacobian Approximations with Optimal Hereditary Properties , 1997 .

[33]  Lorenz T. Biegler,et al.  Large-Scale Optimization with Applications : Part II: Optimal Design and Control , 1997 .

[34]  Nicholas I. M. Gould,et al.  Trust Region Methods , 2000, MOS-SIAM Series on Optimization.

[35]  Sven Leyffer,et al.  Nonlinear programming without a penalty function , 2002, Math. Program..

[36]  Sven Leyffer,et al.  On the Global Convergence of a Filter--SQP Algorithm , 2002, SIAM J. Optim..

[37]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[38]  Nicholas I. M. Gould,et al.  Global Convergence of a Trust-Region SQP-Filter Algorithm for General Nonlinear Programming , 2002, SIAM J. Optim..

[39]  R. Fletcher,et al.  Filter-type Algorithms for Solving Systems of Algebraic Equations and Inequalities , 2003 .

[40]  Roger Fletcher,et al.  On the global convergence of an SLP–filter algorithm that takes EQP steps , 2003, Math. Program..

[41]  J. Nocedal,et al.  An active-set algorithm for nonlinear programming using linear programming and equality constained subproblems , 2003 .

[42]  Clóvis C. Gonzaga,et al.  A Globally Convergent Filter Method for Nonlinear Programming , 2003, SIAM J. Optim..

[43]  Nicholas I. M. Gould,et al.  An algorithm for nonlinear optimization using linear programming and equality constrained subproblems , 2004, Math. Program..

[44]  Lorenz T. Biegler,et al.  Line Search Filter Methods for Nonlinear Programming: Motivation and Global Convergence , 2005, SIAM J. Optim..

[45]  Roger Fletcher.  A New Low Rank Quasi-Newton Update Scheme for Nonlinear Programming , 2005, System Modelling and Optimization.

[46]  Lorenz T. Biegler,et al.  Line Search Filter Methods for Nonlinear Programming: Local Convergence , 2005, SIAM J. Optim..

[47]  P. Toint,et al.  Global Convergence of a Non-monotone Trust-Region Filter Algorithm for Nonlinear Programming , 2006 .

[48]  P. Toint,et al.  A brief history of filter methods. , 2007 .

[49]  Clóvis C. Gonzaga,et al.  Global Convergence of Filter Methods for Nonlinear Programming , 2008, SIAM J. Optim..

[50]  Panos M. Pardalos,et al.  High Performance Algorithms and Software in Nonlinear Optimization , 2011 .

[51]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .